On Hamilton cycles in Erd\H{o}s-R\'{e}nyi subgraphs of large graphs

TL;DR

This paper investigates the emergence of Hamilton cycles in Erdős-Rényi subgraphs of large graphs with high minimum degree, establishing a threshold for Hamiltonicity based on edge retention probability.

Contribution

It introduces a threshold function for Hamiltonicity in Erdős-Rényi subgraphs of large graphs with high minimum degree, linking Hamiltonian properties to edge retention probability.

Findings

Hamilton cycles appear when minimum degree reaches two.

Threshold function for Hamiltonicity in Erdős-Rényi subgraphs.

High probability results for Hamiltonian properties.

Abstract

Given a graph $\Gamma = (V, E)$ on $n$ vertices and $m$ edges, we define the Erd\H{o}s-R\'{e}nyi graph process with host $\Gamma$ as follows. A permutation $e_1,\dots,e_m$ of $E$ is chosen uniformly at random, and for $t\leq m$ we let $\Gamma_t = (V, \{e_1,\dots,e_t\})$. Suppose the minimum degree of $\Gamma$ is $\delta(\Gamma) \geq (1/2 + \varepsilon)n$ for some constant $\varepsilon > 0$. Then with high probability, $\Gamma_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two. Given $0\leq p\leq 1$ we let $\Gamma_p$ be the Erd\H{o}s-R\'{e}nyi subgraph of $\Gamma$, obtained by retaining each edge independently with probability $p$. When $\delta(\Gamma)\geq (1/2 + \varepsilon)n$, we provide a threshold function $p_0$ for Hamiltonicity, such that if $(p-p_0)n\to -\infty$ then $\Gamma_p$ is not Hamiltonian whp, and if $(p-p_0)n\to\infty$ then $\Gamma_p$ is Hamiltonian whp.